Search any question & find its solution
Question:
Answered & Verified by Expert
The shortest distance between the skew-lines $\mathbf{r}=(-\hat{\mathbf{i}}+3 \hat{\mathbf{k}})+t(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})$ and $\mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+s(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ is
Options:
Solution:
2698 Upvotes
Verified Answer
The correct answer is:
$\frac{22}{\sqrt{17}}$
Given, lines $\mathbf{r}=(-\hat{\mathbf{i}}+3 \hat{\mathbf{k}})+t(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})$
and
$\mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+s(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$
Shortest distance between lines is
$\begin{aligned}
& \mathrm{SD}=\frac{((3+1) \hat{\mathbf{i}}+\hat{\mathbf{j}}-(\hat{\mathbf{k}}+3) \hat{\mathbf{k}})-[(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})}{\times(2 \mathbf{i}-\mathbf{j}+2 \mathbf{k})]} \\
& \mathrm{SD}=\frac{(4 \hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{i}}) \cdot(12 \hat{\mathbf{j}}+6 \hat{\mathbf{i}})|\times|(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}) \mid}{|12 \hat{\mathbf{i}}+8 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}|} \\
& \mathrm{SD}=\frac{48+8+32}{\sqrt{144+64+64}}=\frac{88}{4 \sqrt{17}}=\frac{22}{\sqrt{17}}
\end{aligned}$
and
$\mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+s(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$
Shortest distance between lines is
$\begin{aligned}
& \mathrm{SD}=\frac{((3+1) \hat{\mathbf{i}}+\hat{\mathbf{j}}-(\hat{\mathbf{k}}+3) \hat{\mathbf{k}})-[(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})}{\times(2 \mathbf{i}-\mathbf{j}+2 \mathbf{k})]} \\
& \mathrm{SD}=\frac{(4 \hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{i}}) \cdot(12 \hat{\mathbf{j}}+6 \hat{\mathbf{i}})|\times|(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}) \mid}{|12 \hat{\mathbf{i}}+8 \hat{\mathbf{j}}-8 \hat{\mathbf{k}}|} \\
& \mathrm{SD}=\frac{48+8+32}{\sqrt{144+64+64}}=\frac{88}{4 \sqrt{17}}=\frac{22}{\sqrt{17}}
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.